Backward Euler Method

An example of an implicit method is the backward Euler method:

$\displaystyle \underline{\hat{x}}(n) \isdefs \underline{\hat{x}}(n-1) + T\dot{\...
...nderline{\hat{x}}(n-1) + Tf[n,\underline{\hat{x}}(n),\underline{u}(n)] \protect$ (8.11)

Because the derivative is now evaluated at time $ n$ instead of $ n-1$, the backward Euler method is implicit. Notice, however, that if time were reversed, it would become explicit; in other words, backward Euler is implicit in forward time and explicit in reverse time.

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Forward Euler Method